An internal combustion engine in which an NO.sub.x absorber which absorbs the NO.sub.x when an air-fuel ratio of an inflowing exhaust gas is lean while releases the absorbed NO.sub.x when the air-fuel ratio of the inflowing exhaust gas is a stoichiometric air-fuel ratio or rich is arranged in an engine exhaust passage, the NO.sub.x generated when burning a lean air-fuel mixture is absorbed by the NO.sub.x absorber, the air-fuel ratio of the air-fuel mixture flowing into the NO.sub.x absorber is switched from lean to the stoichiometric air-fuel ratio or rich for a predetermined constant time before the NO.sub.x absorbing capability of the NO.sub.x absorber is saturated to make the NO.sub.x absorber release the NO.sub.x, and, at the same time, the released NO.sub.x is reduced has been already proposed by the present applicant (refer to PCT/JP92/01279).
However, the releasing speed and the amount of the NO.sub.x released from the NO.sub.x absorber when the air-fuel ratio of the exhaust gas flowing into the NO.sub.x absorber is switched from lean to the stoichiometric air-fuel ratio or rich so as to release the NO.sub.x from the NO.sub.x absorber greatly fluctuate according to the amount of NO.sub.x absorbed in the NO.sub.x absorber, the temperature of the NO.sub.x absorber, and the degree of richness at the time when the air-fuel ratio is switched from lean to rich. Accordingly, when the air-fuel ratio of the exhaust gas flowing into the NO.sub.x absorber is designed to be switched from lean to the stoichiometric air-fuel ratio or rich exactly for a predetermined constant time when the NO.sub.x should be released from the NO.sub.x absorber as in the above-mentioned internal combustion engine, if this constant time is too short, the NO.sub.x absorbed in the NO.sub.x absorber is not sufficiently released, and therefore the NO.sub.x absorbing capability of the NO.sub.x absorber is gradually decreased and the absorber finally becomes unable to absorb the NO.sub.x, while if this constant time is too long, even if the releasing action of the NO.sub.x from the NO.sub.x absorber is completed, for example the air-fuel ratio of the exhaust gas is maintained at rich, and thus a large amount of unburnt HC and CO will be released into the atmosphere.
To prevent a situation where the NO.sub.x absorber becomes unable to absorb the NO.sub.x or a large amount of unburnt HC and CO is released into the atmosphere, it is sufficient if the air-fuel ratio of the exhaust gas flowing into the NO.sub.x absorber is maintained at the stoichiometric air-fuel ratio or rich until the releasing action of NO.sub.x from the NO.sub.x absorber is completed. For this purpose, it becomes necessary to detect when the NO.sub.x releasing action from the NO.sub.x absorber has been completed.
Also, since sulfur is contained in the fuel and lubricating oil of an engine, SO.sub.x is discharged from the engine. This SO.sub.x is also absorbed together with NO.sub.x into the NO.sub.x absorber. However, when the amount of SO.sub.x absorbed in the NO.sub.x absorber is increased, the amount of NO.sub.x that can be absorbed by the NO.sub.x absorber is gradually lowered, and the time from when the releasing of NO.sub.x is started to when the releasing of NO.sub.x is completed gradually becomes shorter. In this case, the reduction of the amount of NO.sub.x that can be absorbed by the NO.sub.x absorber means that the NO.sub.x absorber is deteriorated, and accordingly as the NO.sub.x absorber is deteriorated, the time from when the releasing of NO.sub.x is started to when the releasing of NO.sub.x is completed becomes shorter. Accordingly, also in this case, if it can be detected when the NO.sub.x releasing action from the NO.sub.x absorber is completed, the degree of deterioration of the NO.sub.x can be determined.
In this way, there are various merits in use if it can be detected when the NO.sub.x releasing action from the NO.sub.x absorber is completed, but in actual circumstances, it cannot be detected when the NO.sub.x releasing action from the NO.sub.x absorber is completed.